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Definitions and Examples of Inverse and Ill-Posed Problems c de Gruyter 2008 J. Inv. Ill-Posed Problems 16 (2008), 317–357 DOI 10.1515 / JIIP.2008.069 Definitions and examples of inverse and ill-posed problems S. I. Kabanikhin Survey paper Abstract. The terms “inverse problems” and “ill-posed problems” have been steadily and surely gaining popularity in modern science since the middle of the 20th century. A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas, etc. In this paper we consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A. N. Tikhonov, V. K. Ivanov and M. M. Lavrentiev. Key words. Inverse and ill-posed problems, regularization. AMS classification. 65J20, 65J10, 65M32. 1. Introduction. 2. Classification of inverse problems. 3. Examples of inverse and ill-posed problems. 4. Some first results in ill-posed problems theory. 5. Brief survey of definitions of inverse problems. 1. Introduction Everything has been said before, but since nobody listens we have to keep going back and beginning all over again. Andre Gide First publications on inverse and ill-posed problems date back to the first half of the 20th century. Their subjects were related to physics (inverse problems of quantum was partly supported by Russian Foundation for Basic Research (grants 08-01-00312-a, 08-01-90260-Uzb). 318 S. I. Kabanikhin scattering theory), geophysics (inverse problems of electrical prospecting, seismology, and potential theory), astronomy, and other areas of science. Since the advent of powerful computers, the area of application for the theory of inverse and ill-posed problems has extended to almost all fields of science that use mathematical methods. In direct problems of mathematical physics, researchers try to find exact or approximate functions that describe various physical phenomena such as the propagation of sound, heat, seismic waves, electromagnetic waves, etc. In these problems, the media properties (expressed by the equation coefficients) and the ini- tial state of the process under study (in the nonstationary case) or its properties on the boundary (in the case of a bounded domain and/or in the stationary case) are assumed to be known. However, it is precisely the media properties that are often unknown. This leads to inverse problems, in which it is required to determine the equation coefficients from the information about the solution of the direct problem. Most of these prob- lems are ill-posed (unstable with respect to measurement errors). At the same time, the unknown equation coefficients usually represent important media properties such as density, electrical conductivity, heat conductivity, etc. Solving inverse problems can also help to determine the location, shape, and structure of intrusions, defects, sources (of heat, waves, potential difference, pollution), and so on. Given such a wide variety of applications, it is no surprise that the theory of inverse and ill-posed problems has be- come one of the most rapidly developing areas of modern science since its emergence. Today it is almost impossible to estimate the total number of scientific publications that directly or indirectly deal with inverse and ill-posed problems. However, since the theory is relatively young, there is a shortage of textbooks on the subject. This is under- standable, since many terms are still not well-established, many important results are still being discussed and attempts are being made to improve them. New approaches, concepts, and theorems are constantly emerging. 2. Classification of inverse problems One calls two problems inverse to each other if the formulation of one problem involves the other one. J. B. Keller In our everyday life we are constantly dealing with inverse and ill-posed problems and, given good mental and physical health, we are usually quick and effective in solv- ing them. For example, consider our visual perception. It is known that our eyes are able to perceive visual information from only a limited number of points in the world around us at any given moment. Then why do we have an impression that we are able to see everything around? The reason is that our brain, like a personal computer, com- pletes the perceived image by interpolating and extrapolating the data received from the identified points. Clearly, the true image of a scene (generally, a three-dimensional color scene) can be adequately reconstructed from several points only if the image is familiar to us, i.e., if we previously saw and sometimes even touched most of the ob- jects in it. Thus, although the problem of reconstructing the image of an object and its surroundings is ill-posed (i.e., there is no uniqueness or stability of solutions), our Inverse and ill-posed problems 319 brain is capable of solving it rather quickly. This is due to the brain’s ability to use its extensive previous experience (a priori information). A quick glance at a person is enough to determine if he or she is a child or a senior, but it is usually not enough to determine the person’s age with an error of at most five years. Attempting to understand a substantially complex phenomenon and solve a problem such that the probability of error is high, we usually arrive at an unstable (ill-posed) problem. Ill-posed problems are ubiquitous in our daily lives. Indeed, everyone real- izes how easy it is to make a mistake when reconstructing the events of the past from a number of facts of the present (for example, to reconstruct a crime scene based on the existing direct and indirect evidence, determine the cause of a disease based on the results of a medical examination, and so on). The same is true for tasks that in- volve predicting the future (predicting a natural disaster or simply producing a one week weather forecast) or “reaching into” inaccessible zones to explore their struc- ture (subsurface exploration in geophysics or examining a patient’s brain using NMR tomography). Almost every attempt to expand the boundaries of visual, aural, and other types of perception leads to ill-posed problems. What are inverse and ill-posed problems? While there is no universal formal def- inition for inverse problems, an “ill-posed problem” is a problem that either has no solutions in the desired class, or has many (two or more) solutions, or the solution pro- cedure is unstable (i.e., arbitrarily small errors in the measurement data may lead to indefinitely large errors in the solutions). Most difficulties in solving ill-posed prob- lems are caused by the solution instability. Therefore, the term “ill-posed problems” is often used for unstable problems. To define various classes of inverse problems, we should first define a direct (for- ward) problem. Indeed, something “inverse” must be the opposite of something “di- rect”. For example, consider problems of mathematical physics. In mathematical physics, a direct problem is usually a problem of modeling some physical fields, processes, or phenomena (electromagnetic, acoustic, seismic, heat, etc.). The purpose of solving a direct problem is to find a function that describes a physical field or process at any point of a given domain at any instant of time (if the field is nonstationary). The formulation of a direct problem includes • the domain in which the process is studied; • the equation that describes the process; • the initial conditions (if the process is nonstationary); • the conditions on the boundary of the domain. For example, we can formulate the following direct initial-boundary value problem for the acoustic equation: In the domain n Ω ⊂ R with boundary Γ = ∂Ω, (2.1) it is required to find a solution u(x, t) to the acoustic equation −2 c (x)utt = ∆u − ∇ ln ρ(x) · ∇u + h(x, t) (2.2) 320 S. I. Kabanikhin that satisfies the initial conditions u(x, 0) = ϕ(x), ut(x, 0) = ψ(x) (2.3) and the boundary conditions ∂u = g(x, t). (2.4) ∂n Γ Here u(x, t) is the acoustic (exceeded) pressure, c(x) is the speed of sound in the medium, ρ(x) is the density of the medium, and h(x, t) is the source function. Like most direct problems of mathematical physics, this problem is well-posed, which means that it has a unique solution and is stable with respect to small perturbations in the data. The following is given in the direct problem (2.1)–(2.4): the domain Ω, the coefficients c(x) and ρ(x), the source function h(x, t) in the equation (2.2), the initial conditions ϕ(x) and ψ(x) in (2.3), and the boundary conditions g(x, t) in (2.4). In the inverse problem, aside from u(x, t), the unknown functions include some of the functions occurring in the formulation of the direct problem. These unknowns are called the solution to the inverse problem. In order to find the unknowns, the equations (2.2)–(2.4) are supplied with some additional information about the solution to the direct problem.
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